Unformatted text preview: STABILITY Math 21b, O. Knill HOMEWORK: 7.6 8,20,42,38*,46*, 8.1 10,24,6*,56* Due: Wednesday or Monday after Thanksgiving. LINEAR DYNAMICAL SYSTEM. A linear map x 7→ Ax defines a dynamical system . Iterating the map produces an orbit x , x 1 = Ax, x 2 = A 2 = AAx, ... . The vector x n = A n x describes the situation of the system at time n . Where does x n go when time evolves? Can one describe what happens asymptotically when time n goes to infinity? In the case of the Fibonacci sequence x n which gives the number of rabbits in a rabbit population at time n , the population grows essentially exponentially. Such a behavior would be called unstable . On the other hand, if A is a rotation, then A n ~v stays bounded which is a type of stability . If A is a dilation with a dilation factor < 1, then A n ~v → 0 for all ~v , a thing which we will call asymptotic stability . The next pictures show experiments with some orbits A n ~v with different matrices. . 99 1 1 stable (not asymptotic) . 54 1 . 95 asymptotic stable . 99 1 . 99 asymptotic stable . 54 1 1 . 01 unstable 2 . 5 1 1 unstable 1 . 1 1 unstable ASYMPTOTIC STABILITY. The origin ~ 0 is invariant under a linear map T ( ~x ) = A~x . It is called asymptot ically stable if A n ( ~x ) → ~ 0 for all ~x ∈ IR n . EXAMPLE. Let A = p q q p be a dilation rotation matrix. Because multiplication wich such a matrix is analogue to the multiplication with a complex number z = p + iq , the matrix A n corresponds to a multiplication with ( p + iq ) n . Since  ( p + iq )  n =  p + iq  n , the origin is asymptotically stable if and only if...
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.
 Spring '03
 JUDSON
 Math, Linear Algebra, Algebra

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